From fully quantum thermodynamical identities to a second law equality

Alvaro Alhambra, Lluis Masanes, Jonathan Oppenheim, Chris Perry

Fluctuating StatesPhys. Rev. X 6, 041016 (2016)

Fluctuating WorkPhys. Rev. X 6, 041017 (2016)

Information

TheoryPhysics

Lord Spekkens

Quantum Information Thermodynamics

stochasticthermodynamics

open systems

majorizationtheory

optimal control

finite timethermodynamics

heat devices

foundations of statistical mechanics

MaxwellSzilardLandauerBennett

4

W=kTlog2

Thermodynamics is an information theory

R L

What do we mean by W=kTlog2?(consider the limit of perfect erasure)

A)W=kTlog2 on average, but there will be fluctuations around this value.

B)We can achieve perfect erasure.

C)By using slightly more work on average, you can sometimes gain work when you erase.

D)None of these statements are true.

E)This quiz is undecidable.

What do we mean by W=kTlog2?(consider the limit of perfect erasure)

A)W=kTlog2 on average, but there will be fluctuations around this value.

B)We can achieve perfect erasure.

C)By using slightly more work on average, you can sometimes gain work when you erase.

D)None of these statements are true.

E)This quiz is undecidable.

What do we mean by W=kTlog2?(consider the limit of perfect erasure)

A)W=kTlog2 on average, but there will be fluctuations around this value.

B)We can achieve perfect erasure.

C)By using slightly more work on average, you can sometimes gain work when you erase.

D)None of these statements are true.

E)This quiz is undecidable.

Fluctuating work in erasure

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1

∑s ' , w

P(s ' , w∣s)=1

Main Results

● QIT strengthening of Stochastic Thermodynamics

● Generalisations of doubly-stochastic maps, majorisation

● Second law of thermodynamics as an equality (fine grained free energy)

● Fully quantum identity Stochastic Thermodynamics

● Fluctuations of work and of states

● Proof and quantification of third law of thermodynamics

Outline

● Review of thermodynamics (Macroscopic, QIT)

● Equalities for work fluctuations

● Quantum identities

● Outlook

Outline

● Review of thermodynamics (Macroscopic, QIT)

● Equalities for work fluctuations

● Quantum identities

● Outlook

3 laws of thermodynamics

0) If R1 is in equilibrium with R2 and R3 then R2 is in equilibrium with R3

1) dE = dQ – dW (energy conservation)

2) Heat can never pass from a colder body to a warmer body without some other change occuring. – Clausius

3) One can never attain T=0 in a finite number of steps

The second law

Heat can never pass from a colder body to a warmer body without some other change occurring – Clausius

The second law

In any cyclic process⟨W ⟩≤Δ F

Free Energy

F = <E> – TS<W> rev = F(ρ initial) – F(ρ final )ρ initial → ρ final iff <W>≤ΔF

Free Energy

F = <E> – TS<W> rev = F(ρ initial) – F(ρ final )ρ initial → ρ final iff <W>≤ΔF This is just the first order

term of an equality!

Resource Theories

lass of operations

In reversible theories and under minor assumptions, relative entropy distance to free states is the unique measure of the resource(Horodecki et. al. 2011)

In thermodynamics, will turn out to be the Gibbs state and the measure is

But thermodynamics can also be irreversible

What is Thermodynamics??

A resource theoretic approach: Γ• (ρs, Hs) • adding free states ρB, HB

• work system ρW, Hw

• energy conserving unitaries U (1st law) [U, Hs + Hw + HB ] = 0

• tracing out • can allow changing Hamiltonian by adding switch bit• translation invariant on W: [U,ΔW]=0 [ΔW,Hw]=i

Streater (1995)Janzig et. al. (2000)Horodecki, JO (2011)Skrzypczyk et. al. (2013)Brandao et. al. (2015)

Work

or in the micro - regime

Broadest definition of thermo

Includes other paradigms (Brandao et. al. 2011)

• Hint

• H(t)

• implicit battery: arbitrary U, and take W=trHρ-trHUρU†

• Implemented using very crude control (Perry et. al. 2016)

• c.f. catalytic transformations (Brandao et. al. 2015)

What is the cost of a state transformation? (2nd law)

What do we mean by work?

•No work: Ruch, Mead (1975); Janzig (2000); Horodecki et. al. (2003); Horodecki, JO (2011)

•Deterministic or worst case work: Dahlsten et al. (2010); Del Rio et. al. (2011); Horodecki, JO (2011); Aaberg (2011); Faist et. al. (2013), Egloff (2015)

•Average work: Brandao et. al. (2011); Skrzypczyk et. al. (2013); Korzekwa et.

al (2015)

•Fluctuating work: Jarzynski (1997); Crooks (1999); Tasaki (1999)

Outline

● Review of thermodynamics (Macroscopic, QIT)

● Equalities for work fluctuations

● Quantum identities

● Outlook

When can we go from ρ to σ?(2nd law)

Many Second Laws

H=0 (Noisy Operations), no workmajorisation

ρ→σ iff ρ≻σ ∑k

p(k )≥∑k

q (k )∀ kp(1)≥ p(2)≥p(3) ...

Horodecki et. al. 2003

(Thermal Operations), deterministic work

thermo-majorisation

(β-ordering)

Horodecki, JO (2011)Ruch, Mead, (75)

Fluctuating work

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1

f s :=E s+Tlog p(s)

F=⟨ f s ⟩

=⟨E ⟩−TS

⟨eβ(f s '− f s+w)⟩=1 2nd law equality

∑s ' , w

P(s ' , w∣s)=1

Classical derivation: Seifert (2012)

Corrections to second law

f s :=E s+TlogP (s)

F=⟨ f s ⟩

=⟨E ⟩−TS

⟨eβ(f s '− f s+w)⟩=1

Standard 2nd law⟨ f s '−f s+w ⟩≤0 W≤Δ F

∑k=1

Nβ

k

k !⟨( f s '− f s+w)

k⟩≤0

Fluctuating work in erasure

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1

Same considerations apply to non-deterministic case

What do we mean by W=kTlog2?(consider the limit of perfect erasure)

A)W=kTlog2 on average, but there will be fluctuations around this value.

B)We can achieve perfect erasure.

C)By using slightly more work on average, you can sometimes gain work when you erase.

D)None of these statements are true.

E)This quiz is undecidable.

no work to fluctuating work

doubly stochastic maps majorisation

Gibbs-stochastic maps thermo-majorisation

fluctuating work linear program

∑s

P(s '∣s)=1

∑s '

P(s ’∣s)=1

∑s

P(s '∣s)eβ(Es'−Es)=1

∑s '

P(s ’∣s)=1

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1

∑s '

P(s ’∣s)=1

Thermo-majorisation curves

Outline

● Review of thermodynamics (Macroscopic, QIT)

● Equalities for work fluctuations

● Quantum identities

● Outlook

Quantum identity

trW [F H 'S+HWΓSW F H S+HW

−1]1S⊗ρW=1S F H (ρ):=e

β

2H

ρ eβ

2H

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1 For classical states

ΓSW=trBU ρSWBU†

c.f. Aaberg (2016)

Quantum identity

Masanes, JO (2014)

Generalised Gibbs-stochastic

Second Law equality

Generalised Jarzynski equality

Crooks equation1

(Petz recovery)

Fully quantum identities

1 c.f. Åberg (2016)

Generalised Gibbs-stochastic

Second Law equality

Generalised Jarzynski equality

Crooks equation(Petz recovery)

Quantum identities w/ diagonal input

c.f. Sagawa, Ueda (2011)Schumacher (2014)Manzano (2015)

∑s , w

P(s ' ,w∣s)eβ(Es '−Es+w )=1

⟨eβ(f s '− f s+w)⟩=1

⟨eβ(w− f s)⟩=Z 'S

Outline

● Review of thermodynamics (Macroscopic, QIT)

● Equalities for work fluctuations

● Quantum identities

● Outlook

Outlook and open questions● Fluctuations of states

● Probabilistic transformation (LOCC: Vidal 1999, Jonathan & Plenio 1999), Alhambra et. al. (2014); Renes (2015); Narasimhachar, Gour (2016)

● Generalised third laws● Reeb, Wolf (2013)

● Thermal machines: Masanes, JO (2014)

● Fully quantum fluctuations (non-commuting case)● Recovery Maps

● Alhambra et. al. (2015), Åberg (2016), Alhambra et.al. (2016)

● Embezzlement of work? (Brandao et. al. 2015)

● Autonomous machines and clocks● Horodecki et. al. (2011), Woods et. al. (2016)

*

ρ σ

Probabilistic state transformations

W

ρ ρ'= p*σ+(1-p*)X

V l(ρ)=∑s=1

l

p(s)

2W ρ⇒σ≤ p≤2−Wσ ⇒ρ

c.f. entanglement theoryG. Vidal (1999)Jonathan, Plenio (1999)

Probabilistic state transformations

*2W ρ⇒σ≤ p≤2−Wσ ⇒ρ

*

Probabilistic state transformations

∑s

T (s '∣s)e−β Es≤e−βEs '

∑s

T (s '∣s) p(s)≥p p(s ' ) Renes (2015)

Quantitative third law(Masanes, JO; to appear in Nat. Comm.)

T '≥αT

t 2d+1

Heat Theorem (Planck 1911): when the temperature of a pure substance approaches absolute zero, its entropy approaches zero

Unattainability Principle (Nernst 1912): any thermodynamical process cannot attain absolute zero in a finite number of steps or within a finite time

Thermal Machines

● Like Turing Machines

● In a finite time, they interact with a finite volume and inject a finite amount of work

●

● Bath of volume V has sub-exponential density of states Ω(E)

t≥1vV 1/d t≥

1uwmax

Outlook and open questions● Fluctuations of states

● Probabilistic transformation (LOCC: Vidal 1999, Jonathan & Plenio 1999), Alhambra et. al. (2014); Renes (2015); Narasimhachar, Gour (2016)

● Generalised third laws● Reeb, Wolf (2013)

● Thermal machines: Masanes, JO (2014)

● Fully quantum fluctuations (non-commuting case)● Recovery Maps

● Alhambra et. al. (2015), Åberg (2016), Alhambra et.al. (2016)

● Embezzlement of work? (Brandao et. al. 2015)

● Autonomous machines and clocks● Horodecki et. al. (2011), Woods et. al. (2016)